The speaker has come to the lecture in order to learn about the stars from an expert. The astronomer has knowledge – lots and lots of it, in fact – but is he wise?
When the proofs, the figures, were ranged in columns before me; (line 2)
The astronomer's perspective is limited to abstractions and generalities. A number is the ultimate abstraction. There is no number "2" in the world – there's only two of this, two of that, etc. The astronomer has organized his figures into complicated arrangements, but how are we supposed to use this knowledge? (OK, so once we started launching satellites, astronomy became a lot more useful.)
When I was shown the charts and the diagrams, to add, divide, and measure them; (line 3)
Whitman piles up examples of abstract knowledge throughout the first half of the poem. We don't want to make it sound like we're giving astronomers a bad rap – we're just the messengers! But seriously, interpreting the poem as an attack on math or science seems too narrow to us.
When I, sitting, heard the astronomer, where he lectured with much applause in the lecture-room, (line 4)
In many ways, a "lecture" goes against the spirit of the virtue of self-reliance, most famously described by the American transcendentalist Ralph Waldo Emerson, but also practiced by Whitman. In a lecture, you aren't learning something from direct experience; you're having it transmitted to you passively in words.
Look'd up in perfect silence at the stars. (line 8)
The end of the poem contrasts the speaker's wisdom with the dry knowledge of the astronomer. He discovers the beauty of the stars for himself, on his own terms.